PRIME-PHASE SEQUENCES WITH PERIODIC CORRELATION-PROPERTIES BETTER THAN BINARY SEQUENCES

Let p be an odd prime, n greater-than-or-equal-to 2 be an integer, and omega be a complex primitive pth root of unity. A construction is presented for a family of p(n) p-phase sequences (symbols of the form omega-i), where each sequence has length p(n) - 1, and where the maximum nontrivial correlation value C(max) does not exceed 1 + square-root p(n). A complete distribution of correlation values is provided. As a special case of this construction, a previous construction due to Sidelnikov is obtained. The family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs, the present design has the additional advantage of not requiring an alphabet of size larger than 3. When compared with the (binary) family of Gold sequences, for essentially the same family size, C(max) is smaller by a factor of square-root 2 (3 dB). In relation to the small set of Kasami sequences, for the same maximum correlation value, the size of the present design equals the square of the size of the latter. The new sequences are suitable for achieving code-division multiple-access, and are easily implemented using shift-registers. They were discovered through an application of Deligne's bound on exponential sums of the Weil type in several variables. The sequences are also shown to have strong identification with certain bent functions.